From an IB Economics book (in the microeconomics section):
The part in blue suggests market failure is the same as allocative inefficiency, but the part in red seems to suggest that market failure is one cause or type of allocative inefficiency.
What is the difference between the two, if any?
In this context 'market' usually means some allocation mechanism. For example free market, or a market where someone has monopoly power. Allocation of goods and resources can be done in other ways: e.g. you can divide all resources evenly among production processes and all goods evenly between consumers. This will generally result in allocative inefficiency (were all production processes equally important and efficient?). But it is not market failure because there was no market, just an allocator/social planner.
Allocative inefficiency refers to a situation where good G was bought by buyer B1 ("allocated" to buyer B1) where it would have been more "efficient" if it had been bought by a different buyer B2 ("allocated" to B2 instead of B1). Good G could be producer goods like oil, lumber, plant equipment, or even consumer goods like sandwiches.
In a free market, the only way buyer B1 gets to bid a good G away from buyer B2 is if B1 voluntarily pays more for G than B2 is willing to pay. This means that in a free market G is always allocated "efficiently", because it goes to the highest bidder, ie the bidder who is obtaining the highest "utility" from G at that point in time (that's why he bid highest). In other words, in a free market, goods are always allocated efficiently.
A good G can be allocated inefficiently to B2 instead of B1 if the market is actively prevented by an external force from allowing B1 to bid G away from B2. There are many ways that this happens today:
For example, B2 may be subsidized to buy G, which means money is forcibly taken from some tax payers and given to B2, thereby enhancing B2's ability to bid some units of G away from B1. An example of this is when a solar panel manufacturer is subsidized, thereby enhancing his ability to bid away goods from other manufacturers via this mechanism. The subsidy has forcibly disrupted the market and caused allocative inefficiency, as compared to the efficient allocation that would have been provided by the free market.
As another example, the regulatory burden imposed on B1 may be higher than that imposed on B2, causing B1 to bleed resources away thereby reducing its ability to bid some units of G away from B2. Say a mom & pop restaurant B1 is forced to comply with a newly imposed food safety regulation, and they spend a higher percentage of their revenue on compliance than would a chain restaurant competitor B2 who already has a compliance department and therefore incurs smaller marginal cost of compliance. In this case, mom & pop restaurant B1 bleeds more compliance cost than chain competitor B2, reducing mom & pop B1's ability to bid goods G away from chain B2. In the free market some units of good G would go to B1, but after forcible imposition of the additional regulatory compliance some of those units instead go to B2. The regulatory burden has disrupted the free market and caused allocative inefficiency compared to the efficient allocation that would have taken place in the free market.
This is why the phrase "market failure" is misleading and incorrect. The free market allocates efficiently, because goods go to the highest bidder. It is "forced market disturbance" that causes allocative inefficiency by disrupting the free market. Therefore, more forced market disturbance means less efficientcy, and less forced market disturbance means more efficiency, with maximum allocative efficiency when the market is free of all forced market disturbances (taxes, regulation, subsidies, red tape, etc.)
